Hypoexponential distribution
Encyclopedia
In probability theory
the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory
, teletraffic engineering
and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation
less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution
which has coefficient of variation of one.
. The hypoexponential is a series of k exponential distributions each with their own rate 
, the rate of the 
exponential distribution. If we have k independently distributed exponential random variables 
, then the random variable,
is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of
.
. The phase-type distribution is the time to absorption of a finite state Markov process
. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from state i to i+1 with rate
until state k transitions with rate 
to the absorbing state k+1. This can be written in the form of a subgenerator matrix,
For simplicity denote the above matrix
. If the probability of starting in each of the k states is
then
.
has cumulative distribution function
given by,
and density function,
where
is a column vector of ones of the size k and 
is the matrix exponential
of A. When
for all 
, the density function can be written as
where
are the Lagrange basis polynomials
associated with the points
.
The distribution has Laplace transform of
Which can be used to find moments,
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
, teletraffic engineering
Teletraffic engineering
Telecommunications traffic engineering, teletraffic engineering, or traffic engineering is the application of traffic engineering theory to telecommunications...
and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation
Coefficient of variation
In probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution. It is also known as unitized risk or the variation coefficient. The absolute value of the CV is sometimes known as relative standard deviation , which is...
less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
which has coefficient of variation of one.
Overview
The Erlang distribution is a series of k exponential distributions all with rate. The hypoexponential is a series of k exponential distributions each with their own rate , the rate of the exponential distribution. If we have k independently distributed exponential random variables , then the random variable,is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of
.Relation to the phase-type distribution
As a result of the definition it is easier to consider this distribution as a special case of the phase-type distributionPhase-type distribution
A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random...
. The phase-type distribution is the time to absorption of a finite state Markov process
Markov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from state i to i+1 with rate
until state k transitions with rate to the absorbing state k+1. This can be written in the form of a subgenerator matrix,For simplicity denote the above matrix
. If the probability of starting in each of the k states isthen
.Characterization
A random variable has cumulative distribution functionCumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
given by,
and density function,
where
is a column vector of ones of the size k and is the matrix exponentialMatrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
of A. When
for all , the density function can be written aswhere
are the Lagrange basis polynomialsLagrange polynomial
In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...
associated with the points
.The distribution has Laplace transform of
Which can be used to find moments,
See also
- Exponential distributionExponential distributionIn probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
- Erlang distribution
- Hyper-exponential distribution
- Phase-type distributionPhase-type distributionA phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random...
- Coxian distribution